Problem: Find the units digit of $9^{8^7}$.
Explanation: If $k$ is a positive even integer, then $$9^{k} = 81^{k/2}=\overbrace{(81)(81)\cdots (81)}^{k/2\text{ times}},$$so $9^k$ has a units digit of 1. Since 8 is even, $8^7$ is even.  Therefore, $9^{8^7}$ has a units digit of $\boxed{1}$.